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Daniel L. Jensen and James C. McKeown 


College of Commerce and Business Administration 

University of Illinois at Urbana-Champaign 

College of Commerce and Business Administration 
University of Illinois at Urbana-Champaign 
March 6, 1973 


Daniel L. Jensen and James C. McKeown 

if 9 5 


Daniel L. Jensen and James C. McKeown 
University of Illinois at Urbana-Champaign 

The investigation of cost variances has been the subject of several 
papers in recent years. These papers develop models that lead to a 
decision as to whether or not to investigate a process after a cost 
variance is observed. The investigation is usually conceived as a 
single-stage investigation process with a constant cost; in other words, 

the decision does not allow for a choice among alternative investigative 


Yet such a choice is apt to be necessary in practice. Accounting 
variances arise from a multiplicity of causes. \'Jhen a variance is ob- 
served, it is reasonable to suppose that the structure of any insuing 
investigation will be influenced by available information about the 

likelihood of the various possible causes of the variance and about the 

costs and savings associated with discovery and correction of each 

variance-causing condition. 

The purpose of this paper is to consider a multiple-stage invest- 
igation process in which the decision involves a choice among several 

investigative techniques and a determination of the order of their 

application. Although the complexity of the decision rule increases 

with the complexity of the investigative process, the decision rule may 

be operated by a simple graphical technique. 

The Single-Step I nvest igation 

A process is in one of two states; it is either in control or out 
of control . If the process is in con rol, then corrective action cannot 
improve its operation. If it is out cf control, then corrective action 
will secure certain benefits. Unfortunately j we cannot know vrith certainty 
which state actually exists unless v-je investigate and incur the attendant 
costs. Clearly we do not wish to investigate a process when the cost of 
doing so exceeds the benefits obtained thereby. Since the state of the 
process is uncertain before investigation, we formulate a decision model 
that leads to a choice between investigating and not investigating so as 
to maximize the net expected benefits. 

Prior to the decision, we formulate probabilities about the process 
being in or out of control and about the probability of observing variances 
of different sizes given that the process is in control and given that it 
is out of control. Pq(IC) designates the prior probability that the process 
is in control and Pq(OC) = 1 - Pq(IC) is the probability that the process 
is out of control. P(X;IC) and P(X;OC) designate the conditional probabilities 
of observing a variance X given that the process is in control and given 
that the process is out of control, respectively. 

When a variance, X, occurs, we revise our prior probabilities according 
to Bayes Theorem as follows: 

p ac) = p (ic- X) = p(x? ic) p„(ic) 

^^n''^^'' ^n'^^^* ^^ P(X;IC).Pjj(IC) + P(X;OC).Po(OC) 

P^(OC) = 1 - P^(IC) 
where Pj^(IC) is the revised probability that the process is in control given 
an observed variance X. Notice that three probabilities are required to 
reach the revised est3.mates — the prior probability that the process is in 
control, Pq(IC); the conditional probability of a variance X when the process 

is in control, P(X;IC); and the conditional probability of a variance X 
vhen the process is out of control, P(X;OC). 

Having revised the probability that the process is in control, we 
proceed to calculate the expected value of the actions open to us. In 
the single-stage investigation problem only two actions are available — 
investigate or do not investigate. The loss matrix associated with this 
simple decision is given below: 



In Control 

Out of Control 

Investigate (I) 
Do not Investigate 




The table shows the outlay costs associated with each combination 
of action and state. Investigation incurs a known constant cost, C, and 
an out-of-control process occasions a constant loss, L, which is avoided 
if an investigation is undertaken. The decision maker wishes to choose 
between investigating and not investigating so that he minimizes the 
expected outlay cost. The expected cost of investigation, E(I), is the 
sum of two products: (1) the cost of investigation multiplied by the 
probability that the process is in control and (2) the cost of investigation 
multiplied by the probability that the process is out of control, that is, 

E(I) = C-P(IC) + C-P(OC) 
The expected cost of not investigating, E(NI), is the product of (1) the 
loss and the probability that the process is out of control, that is, 

E(NI) = L*P(OC) 

The decision maker calculates E(X) and E(NI) and chooses the action with 
the smaller expected cost. Alternatively, the decision rule may be formulated 
as a test of the revised probability that the process is in control. The 
expected cost of investigating will be less than the expected cost of not 
investigating only if the revised probability that the process is in control 
is greater than the ratio (C - L) / L. This is the familiar result obtained 
by Bierman, Fouraker and Jaedicke and by Dyckman. 

Multiple-Step Investigations 

The single-step investigation model reviewed above contemplates a fixed 
set of investigative procedures with a constant cost. In practice, however, 
it is unlikely that the search for causes of an observed deviation of variance 
exhibits such simplicity. It is probable that different variance-causing 
conditions call for different investigative procedure investigation and that 
the decision is not merely whether or not to investigate, but also how to 

The development of a multiple-step investigation model requires the 
partitioning of investigative procedures into groups or steps containing 
only inseparable elements — procedures that must be performed together. 
As in the single-step model, if a step is undertaken the investigative 
activity is completely specified, and exactly the same procedures are 
applied regardless of the occasion on which the step is undertaken. It is 
therefore, reasonable to view the cost of each step as more or less constant. 

The multiple-step model considered here constrains the relationship 
between time and conditions that dislodge a process from control. No more 
than one such condition may be operative in any given period. Such a 
constraint simplifies the investigation because once the condition is 

discovered, we need to look no further. On the other hand, if several 
conditions may combine to effect an out-of-control state, then the in- 
vestigation must determine the extent of the variance attributable to a 
particular condition in addition to the presence or absence of the condition* 
Moreover, the presence of a condition alone may not bring loss of control 
although its presence in combination vith other conditions will bring ]^09» 
of control. Cases of this complexity are beyond the scope of thi5 papejr. 

Each stage of investigation is directed toward the discovery of >a 
well-defined condition whose presence creates an out-of-control &i-cuaJ:lon 
and causes a known loss. The outcome of each stage is binary in the sense 
that the condition is found to be present or absent. Moreover, the in- 
vestigative techniques are assumed to be error-free, which precludes an 
erroneous conclusion as to the presence or absence of the condition under 
Investigation. Finally, the corrective procedures which follow discovery 
of such conditions are assumed to be effective in avoiding the known loss. 

The revision of probabilities, after the observation of a variance 
X, in the presence of several out-of -control states, requires an application 
of Bayes theorem. In general if the process is in one of J states, S., we 
revise each of the J prior probabilities, Pq'^S.), according to Bayes Theorem 

as follows: 

P(X;S^)>Po (s,) 

^n(Si) = Pn(Si;X) = J 

Z P(X;S ).Po(S.) 
j=l ^ ^ 

In the cases examined here, three states are possible: (1) an in-control 

state, (IC); (2) an out-of-control state owing to the presence of condition A, 

(OCA); and (3) a second out-of-control state owing to condition B, (OCB) . 

After observing a variance X, the three prior probabilities are revised as 

' y- 


P(X;IC)'P (IC) 

^n^^^^ P(X;IC)Po(IC) + P(X;OCA)Fo(OCA) + P(X;OCB)Po(OCB) 

P„(OCA) = P(X; OCA)-P^(OCA) 

P(X;IC)Pq(IC) + P(X;OCA)P^(OCA) + P(X:OCB)P (OCB) 

Pj^(OCB) = 1 - P„(IC) - I^ (OCA) 

The revised probabilities are conditional on the observation of a 
variance X and require prior knowledge of the conditional density 
functions of X given each of the three possible states as well as 
prior estimates of the probability that each state occurs. 

The Case of Two Investigative Steps Unconstrained 
as to Combination or Order of Application 

The allowable combinations of investigative steps and the order 
in which they are applied niay be constrained in many different ways. 
For example, Dyckman considers a case in which two investigative 
procedures — an exporatory investigation and a full investigation — are 
alternative to one another and may no^ be applied together in any order. 
The case considered here imposes no constraints as to the allowable 
combinations of procedures or as to the order of their application. 

When the two investigative steps may be undertaken in any order, 
the decision entails a choice among five alternative actions: (1) do 
not investigate, (NX); (2) investigate for condition A only, (lA) ; 
(3) investigate for condition B only, (IB); (4) investigate first for 
condition A, then for condition B, (lAB); and (5) investigate first for 
condition B, then for condition A, (IBA) . The loss associated with each 
action under the three possible states of the process is given in Table 1. 

If the objective is to rainimize the expected loss, then the decision 
requires a determination of the expected loss of each action (as given 
in the far right-hand column of Table 1) and a selection of the action 
with the smallest expected loss. 

The selection of the minimum- loss action requires at most ten pair- 
wise comparisons of the expected values of the five alternative actions. 
Each of the ten comparisons may be represented by an inequality that specifies 
the choice of one action over another. For example, the action NX (no 
investigation) is preferred to the action lA (investigate for A only) 
when E(NI) is less than E(IA), that is, when 

Vn^A^ + Vn(B) < C^ + LbPn(B) 
or, equivalently, 

VPn(A) - C^ < 0. 
If the inequality is satisfied, then NI is preferred to lA and lA (and 
the comparisons involving lA) may be eliminated from further consideration. 
If La'Pjj(A) - Cg = 0, then we are indifferent between lA and NI. On the 
other hand, if 1^ ^^^^) ~ C^ > 0, then lA is preferred to NI and NI is 
eliminated from further consideratioas. We then proceed to compare the 
remaining (non-eliminated) pairs of expected values in a similar way. 
Eventually a preferred action will be indicated. Tlie ten pairwise com- 
parisons of expected values are given in Table 2. Although this procedure 
is somewhat tedious it may be simplified by a simple graphical method. 

TABLE 1. — Loss Matrix for Case of Three Investigative Steps 

where the Order of Their Application is Unconstrainted 






Expected Value 




E(NI)=L^P^(A)+L^ P^(B) 





E(IA)= C^ + Lj^ P(B) 





E(IB)= % + L^ P(A) 





E(IAB)= C^+Cj^[l-P^(A)] 





E(IBA)= C^+C^[1-P^(B)] 





TABLE 2. — Inequalities Leading to a Minimum Expected Cost Action 


Inequality Condition 

NX over lA 

^a Pn(A) - C^ < 

NI over IB 

\ Pn<2) - C^ < 

NX over XAB 

<La+Cb) Pn^A>+Vn(B)-Ca-Cb < 

NI over IBA 

L^Pn(A)+(Lb+Ca)P^(B)-Cb-Ca < 

lA over IB 

LbPn(B>-Vn<A)+Ca-Cb < 

XA over XAB 

LbPn(B)+CbPn(A)-C^ < 

lA over XBA 

(V^a^ Pn(B) -Cb<-0 

IB over XAB 

(La+C^) Pn(A) -C^< 

IB over IBA 

LaP(A) + C^PCB) -Ca< 

lAB over IBA 

■ CaP(B) -Cj^P(A)< 


Illustration of Graphical Procedure 
Since the costs and avoidable losses associated with investigation 
are known constants, each pair-wise comparison of expected 
values represents a partition of the probability space which gives all 
possible combinations of P (A) and P„(B). If the space is partitioned 
for each of the ten comparisons, each resulting region of the space 
identifies pairs of P^CA) and PjiCB) that signal a unique least-cost action. 
Once the graph is determined, the choice among the five alternative actions 
requires only the calculation of P^CA) and PqCB) and the location of the 
region on the graph with the pair of probabilities is associated. 

A numerical illustration will clarify the application of the model and 
the graphical procedure. Suppose that condition A leads to a loss of $100 
and Investigation for condition A costs $50; condition B leads to a loss 
of $300 and investigation for B costs $100. The graph in Figure 1 is 
derived from this data by a simple procedure. First the data is substituted 
into each of the ten inequalities given in Table 2 and the corresponding 
boundary lines are drawn on the grap'i. Each inequality in Table 2 gives 
rise to a line of the graph. For example, the line specifying the preference 
relation between the actions lA and IB is given by the equation 300P(B) - 
lOOP(A) - 50 = or, equivalently, 6P(B) - 2P(A) - 1 = 0. Probability 
pairs above this line indicate a preference for action IB over lA while 
pairs below indicate a preference for lA over IB. The other lines are 
obtained from the remaining nine inequalities in a similar way. 

Next, each cell in the resulting partition of the probability space 
is identified with the least-cost action. The performance of this second 
step is straight-forward, but its rationale requires some explanation. 


i .^A 



_ IBA _- 


1.0 -• 




In general, we will not investigate if ^^(A) and Pjj(B) are sufficiently 
snail. Clearly, we will not investigate if P^CA) and Pji(B) are both equal 
to zero which corresponds to the origin. Nor will we investigate in those 
cells whose boundaries include the origin. As we move to other regions 
whose boundaries do not adjoin the origin, we. still do not investigate 
unless a boundary is crossed that signals an investigation of some kind. 
When such a boundary is crossed, all points within the region entered 
signal the kind of Investigation designated as we cross the boundary. In 
general, moving from any region in which the decision is known, to an adjacent 
region, leaves the decision unchanged unless the shared boundary involves 
the decision of the first region; if it does, the decision is changed to 
the other action involved. 

In other words, the second step in generating the graph begins by 
locating the cells that adjoin the origin of the probability space and by 
designating the probability pairs contained therein as signaling no Invest- 
igation. We proceed from these "no investigation" regions to adjacent 
regions and, then, to other adjacent regions until every region of the 
probability space has been identified with a least-cost action according 
to the following rule: as one moves from any region in which the least- 
cost action is known, to an adjoining region, determine whether or not 
the pair of actions associated with the boundary crossed Includes the least- 
cost action of the region from which one moves. If the pair does not include 
the action signaled in the first region, then the least-cost action of the 
second region is the same as that of the first. If the pair does include 
the action signaled by the first region, then the least-cost action of the 
second region is the other member of the pair. In this way every cell of 
the partitioned probability space is identified with a least-cost action. 


After construction of the graph, the decision rule may be implemented 
by merely calculating Pn(A) and P^^^^' Plotting the corresponding point on 
the corresponding point on the graph, and taking the least-cost action identified 
with the region within which the point falls. For example, if the revised 
probabilities, P (A) and P (B), are found to be 0.5 and 0.4, respectively, 
then the action lAB is signaled. 

The complexity of the investigation process has an important impact 
on the decision rule in models for the investigation of cost variances. In 
general, the more complex the representation of the investigation process, 
the more complicated the decision rule. In some instances, complexity is an 
impediment to the application of a model despite the fact that it provides 
a better fit to reality. In the case examined here, however, a simple graphical 
procedure permits a straight-forward application of the model despite the 
complication introduced in the decision rule. 



See, for example, H. Biennan, L. E. Fouraker, and R. K. Jaedicke, 
"A Use of Probability and Statistics in Performance Evaluation," The 
Accounting Review , XXXVI (July, 1961), 409-417; and T. R. Dyckman, "The 
Investigation of Cost Variances," Journal of Accounting Research , VII 
(Autumn, 1969), 215-244. 

^Dyckman (1969) considers an exception in the form of a three-action 
model for the purpose of choosing between an exploratory Investigation, 
a full investigation and no investigation. The full investigation is 
assured of correcting an out-of-control state, if one exists, but the 
exploratory investigation may fail to do so with known probability. This 
analysis differs from the analysis in this paper in that the two Investigative 
procedures are strictly alternative to another and cannot be undertaken 
together. See pp. 228-230. " . . 

The analysis presented here assumes that the cost of correcting 

an out-of-control condition, over and above the cost of the investigation 
leading to its discovery, is negligible. A cost of correction can be in- 
corporated into the analysis without great difficulty. See Harold Bierman, 
Jr. and Thomas R. Dyckman, Managerial Cost Accounting . New York: The 
Macmillan Company, 1971, pp. 33-53. 

Bierman and Dyckman (1971) suggest this among other possible extensions 
of the analysis. See p. 52. 

^Dyckman (1969), pp. 228-230.